Method to reduce sample size in accelerated reliability verification tests

ABSTRACT

A method for determining a sample size W required for accelerated testing of a product includes the steps of selecting a Reliability Goal R as appropriate for the product, selecting a Confidence Level CL appropriate for the accuracy required from the results of the accelerated testing, selecting the number of testing cycles N t  defining the accelerated testing period, calculating the Sample Size W for the accelerated testing as ##EQU1## and then testing the W product samples for the N t  testing cycles to validate the required Reliability when no test failures are observed over the N t  testing cycles. A method for determining the Number of Cycles N t  required for accelerated testing of a product having a service lifetime is also described. In either of the above methods, if any failures are observed in the N t  testing cycles/time, then the number of testing cycles/time may be extended to at least 2 N t  and a new Confidence Level is calculated ##EQU2## The Reliability Goal R for the product design is validated if the new Confidence Levels CL NEW  is greater than a CL min  value specified as a minimum confidence level required for the accelerated testing method.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a statistical method for validating thereliability of a product based upon the number of testing cycles/timeand the sample size in an accelerated test.

2. Description of the Prior Art

New designs and manufacturing processes often are validated forlong-term reliability by accelerated reliability verification tests,such as vibration test and thermal cycle/shock tests. These tests areincluded in many design verification and product validation (DV/PV)processes. The goal of the verification test is to obtain a correctreliability assessment of the products in field operation. Becausefailure is a random phenomenon, the fewer the samples in a test, thelower the confidence in the test results. Inadequate sample sizeselection in a reliability test often leads to misleading results.Therefore, a complete test specification should include a minimum samplesize requirement. Unfortunately, the available DV/PV specifications ofmost products don't specify the sample size requirement due to lack ofmethodology for accelerated testing. This could be one of the reasonswhy some products pass verification testing, but will not perform fortheir full planned life in the field.

An object of the present invention is to define an accelerated testingmethod whereby either a reduced sample size of products may be testedfor a predetermined number of testing cycles, or whereby a standardsample size may be tested for a reduced number of testing cycles. Eitherof these accelerated tests results in the validation of the reliabilityfor the product design with a confidence level exceeding that requiredfor the accelerated testing.

SUMMARY OF THE INVENTION

A method for determining a sample size W required for acceleratedtesting of a product includes the steps of:

(a) selecting a Reliability Goal R as appropriate for the product.

(b) selecting a Confidence Level CL appropriate for the accuracyrequired from the results of the accelerated testing.

(c) selecting the number of testing cycles N_(t) defining theaccelerated testing period,

(d) calculating the Sample Size W for the accelerated testing as##EQU3## where R(N_(t)) is the reliability function. (e) testing the Wproduct samples for the N_(t) testing cycles to validate the requiredReliability when no test failures are observed over the N_(t) testingcycles.

A similar method for determining the Number of Cycles N_(t) required foraccelerated testing of a product includes the steps:

(a) selecting a Reliability Goal R as appropriate for the product.

(b) selecting a Confidence Level CL appropriate for the accuracyrequired from the accelerated testing.

(c) selecting the number of product samples for the Sample Size W,

(d) calculating the Number of Cycles N_(t) for the accelerated testingas ##EQU4## where R⁻¹ is the inverse function of R(n), and (e) testingthe W product samples for the N_(t) testing cycles to validate therequired Reliability when no test failures are observed over the N_(t)testing cycles.

In either oft he above methods, if any failures are observed in theN_(t) testing cycles, then the number of testing cycles may be extendedto at least 2 N_(t) and a new Confidence Level is calculated as:##EQU5## where NF is the number of failed products within 2N_(t) cycles.

The Reliability R for the product design is validated if the newConfidence Levels CL_(NEW) is greater than a CL_(min) value specified asa minimum confidence level required for the accelerated testing method.

BRIEF DESCRIPTION OF THE DRAWINGS

Other objects, features and advantages of the invention will be apparentfrom studying the written descriptions and the drawings in which:

FIG. 1 is a plot showing the S-N fatigue curve and damage curve.

FIG. 2 is a plot of the R-S-N curve with a survival probability of R.

FIG. 3 is a plot of the relationship between the required sample size Wand the reliability goal R for various Confidence Levels CL.

FIG. 4 is a plot of required sample sizes W as a function of the numberof testing cycles N_(t) for Reliability Goals R equal to 99.0%, 99.9%,99.97% and 99.99% for a specific testing example (Example 1).

FIG. 5 is a plot of power spectral density function measured in fieldand that for the accelerated vibration testing of an electronic modulemounting bracket (Example 2).

FIG. 6 is a plot of the required sample size W as a function of testingtime for reliability goals R of 99%, 99.9%, 99.97% and 99.99% for aspecific testing example (Example 2).

DESCRIPTION OF THE PREFERRED EMBODIMENT

VERIFICATION TESTS BASED ON DAMAGE EQUIVALENCE

A physics-of-failure based approach to develop accelerated reliabilityverification tests for automotive components may be based on the damageequivalence technique. According to this approach, the accelerationlevel and the required test cycles/time in a test are determined byequating the damage induced in test and the damage induced in fieldapplication for a desired operational life. Most cycle/time relatedfailures of electronic packages and mechanical components are due to thefailure mechanism of fatigue, and the cumulative damage can be estimatedby: ##EQU6## where S is stress/strain amplitude n is the number oftesting cycles applied at the stress/strain level of S, and C and m arematerial fatigue properties. Equation 4 shows the linear relationshipbetween the cumulative damage and the applied number of testing cycles.Failure occurs when total damage reaches one hundred percent. The numberof cycles to failure is then determined by setting D=100% in Equation 1,which yields: ##EQU7## Plotting Equation 5 on a Log-Log scale gives theS-N fatigue curve as shown by curve 10 in FIG. 1. It can be observedfrom FIG. 1 that the fatigue property m is the inverse slope of the S-Ncurve. A qualification curve can be generated, as shown by curve 20 inFIG. 1, for a material that is not failed, but has a certain percentageof the cumulative damage (D <100%). According to Equation 4, thequalification curve has the same slope as the S-N curve, and thevertical distance between the verification/qualification curve 20 andthe S-N curve 10 at the desired field operation life is the designmargin.

In a laboratory verification test, in order to produce the damage whichis the same as that induced in field operation for a desired operationallife N_(o) (not cycles to failure), the equivalent number of cycles intest N_(E) is determined according to the verification curve: ##EQU8##where S_(t) and S_(o) are equivalent stress/strain amplitudes applied inlaboratory test and field operation, respectively.

For a given level of stress/strain amplitude, even under very carefullycontrolled laboratory testing conditions, significant variation infatigue life data is observed The lognormal or Weibull distribution isassumed to provide a good model of fatigue life. The mean life, μ_(N)and the standard deviation of the life σ_(N) can be estimated from testdata, using statistical methods such as the maximum likelihoodestimation. A dimensionless measure of variability can be defined by thecoefficient of variation (COV), δ_(N) =σ_(N) /μ_(N). In order todescribe the variation of the S-N data, the R-S-N curve with a survivalprobability of R, as shown in FIG. 2, is often used. Therefore, thevertical distance between the verification curve and the R-S-N curve atthe desired field operation life is the design margin with thereliability goal of R.

The problem of determining the sample size for the test may be analyzedas follows. Equation 6 specifies the equivalent number of cycles forproducts tested in the accelerated test condition corresponding to fieldlife requirements. However, the confidence in a verification test isdependent on sample size. In general, the sample size W is related tothe prespecified reliability goal R, confidence level CL, and allowablenumber of failures in the test NF. The confidence level is theprobability that the quantity ##EQU9## (the reliability estimate) isequal to the reliability goal R. According to statistical theory, thisprobability follows the Binomial distribution: ##EQU10##

Due to the large number of serially connected components existing in anautomotive system, the allocated reliability goal for a component mustbe very high. With an acceptable confidence level (80% to 90%, forexample), the high reliability goal yields unreasonably large samplesize requirements. For example, if the reliability goal is R=99.9%, aconfidence level CL=85%, and with only one failure being allowed in anaccelerated verification test, the calculated sample size will beW=3155. This number is unreasonably large for most electronic packagesand mechanical components. It is obvious that a more efficientassumption set must be found if the testing program is to beaccelerated. Experience has shown that it is unwise to compromiseverification testing in a high quality manufacturing environment byreducing the confidence level to be below about 80%. Actually, even withthe confidence level of 70%, the required sample size is still too high(e.g., W=2439).

ACCELERATED TESTING WITH NO FAILURES ALLOWED

If the number of failures in verification testing is reduced to N=0(that is, no failures are allowed in the test), then Equation 7 becomes:

    CL=1-R.sup.W

Therefore, the required sample size can be determined by the followingequation: ##EQU11##

FIG. 3 shows the relationship between the required sample size and thereliability goal with various confidence levels. As shown in FIG. 3,with a reliability goal of R=99.9% and a confidence level of CL=85%, therequired sample size in a verification test with no failures allowedwill be W=1900. Even using a 70% confidence level, a sample size of 1204still is required. These sample sizes are too large to conduct averification test for commercial products, especially for automotiveelectronics and mechanical components. Therefore, an even more efficientassumption set must be developed.

An engineering technique for reducing the sample size may beaccomplished as follows. In reality, product reliability is a functionof the number of operational cycles before failure, for example,

    R=R(n)

where n is the required cycles in verification test, and R(n) is thereliability function. R(n) is a decreasing function of operating cyclesn.

The concept of sample size reduction may be explained as follows. Assumethat the reliability goal R is specified for a desired field life of10,000 cycles and then assume that the equivalent number of test cyclesis calculated to be 1000 based on the damage equivalence technique for agiven accelerated stress level. Then, the reliability requirement for1000 cycles should be R(1000)=R, because the damage induced in 1000cycles of accelerated testing in the laboratory is known to be the sameas that in field operation for 10,000 cycles. According to Equation 10,it is appropriate to consider that achieving the reliability requirementR(2000) at 2000 test cycles is equivalent to achieving R(1000) at 1000test cycles. This is, if products are qualified at 2000 cycles with thereliability requirement of R(2000), then they are qualified at 1000cycles with the reliability of R(1000) for the same stress condition.Since the reliability function is a decreasing function of the number oftesting cycles (for example, R(2000)<R(1000)), the required sample sizefor a test running to 2000 cycles will be smaller than that in the sametest running to 1000 cycles, if we assume the same confidence level.Substituting Equation 10 into Equation 9, the required sample size isdetermined as a function of the test cycles for the case where nofailures are allowed: ##EQU12##

Actually, an alternative technique could involve an increase in thestress amplitude level. However, any increase in the stress amplitudelevel from an already accelerated level takes the risk of shiftingfailure mechanisms, for example, from high cycle fatigue to low cyclefatigue. Therefore, the approach of increasing test cycle/time to reducesample size is appropriate. Based upon this set of assumptions, aprocess for determining the sample size W required for acceleratedtesting may be expressed as follows:

(a) The Reliability Goal (R) is specified for the product (or component)as appropriate based on the reliability allocation from the systemreliability goal over the product service lifetime. A typicalreliability goal of a product could be 99.99%, but this variable may berange from about 99% to 99.99999%.

(b) The Confidence Level (CL) appropriate for the accuracy of thecalculation/answers required from the accelerated testing is specified.A typical CL is 85%, but this variable may range from about 70% to 95%depending on the critical nature of the application and the importanceof the testing results to the survival of the product being tested.

(c) The Number of cycles N_(t) (duration) to be included in theaccelerated testing is specified. This variable may be selected based onthe simultaneous availability of testing equipment for parallel testing,the availability of product samples to be tested, the acceptable elapsedtime for all accelerated testing, etc. For example, if only a fewtesting systems are available, then the number of cycles will bedetermined largely by the availability of product samples to be testedand the acceptable elapsed time for all accelerated testing. Limitedavailability of product testing samples or long testing cycles mayrequire different assumptions. N_(t) may be measured in the number ofcycles (e.g., power or temperature cycles) or continuous testing time(hours, days, weeks) where cyclic testing is not employed (e.g.,vibration time in hours).

(d) The Sample Size (W) is calculated using Equation (11), which is##EQU13## (e) The W product samples are tested for the specified numberof testing cycles N_(t). If no failures are encountered in the W productsamples tested, then the required Reliability has been validated and nofurther testing is required for the assumed values of CL and R. If oneor more failures are encountered during the testing, then either theproduct must be redesigned to avoid the demonstrated failure, or furthertesting may be required.

Based upon this same set of assumptions, a process for determining thenumber of cycles N_(t) required for accelerated testing max, beexpressed as follows:

(a) The Reliability Goal (R) is specified for the product as appropriatebased on the reliability allocation from the system reliability goalover the product service lifetime. A typical reliability goal of aproduct could be 99.99%, but this variable may be range from about 99%to 99.99999%.

(b) The Confidence Level (CL) appropriate for the accuracy of thecalculation/answers required from the accelerated testing is specified.A typical CL is 85%, but this variable may range from about 70% to 95%depending on the critical nature of the application and the importanceof the testing results to the survival of the product being tested.

(c) The number of product samples available for the sample size W of theaccelerated testing program is specified. This variable may be selectedbased on the number of product samples available (such as when only afew prototypes are available for testing), the availability of testingequipment, the acceptable elapsed time for all accelerated testing,etc., as discussed in more detail with respect to the previous process.

(d) The number of cycles N_(t) required for accelerated testing iscalculated using the following equation which is derived from Equation(11): ##EQU14## where R⁻¹ is the inverse function of R(n). (e) The Wproduct samples are tested for the specified number of testing cyclesN_(t). If no failures are encountered in the W product samples tested,then the required Reliability has been validated and no further testingis required for the assumed values of CL and R. If one or more failuresare encountered during the testing, then either the product must beredesigned to avoid the determined failure, or further testing may berequired.

Using this approach and assumption set, two examples will be providedfor demonstrating the technique of sample size determination. Oneexample involves the thermal cycle testing of leadless solder jointsassociated with chip resistors in electronics modules. A second exampleinvolves the random vibration test for electronic module mountingbrackets.

EXAMPLE 1 SOLDER JOINTS WITH CHIP RESISTORS IN ELECTRONICS MODULES

The solder joints with chip resistors are a significant concern forreliability of electronic modules. For leadless solder joints in anelectronic module operated in the passenger/luggage compartment of anautomotive vehicle, the effective temperature range is approximately 43°C. If the reliability verification test is conducted at a temperaturerange from -30° C. to 100° C., the equivalent number of testing cyclesis 844, which corresponds to 7300 cycles of operational life. Thereliability function of leadless solder joints often follows themodified Weibull distribution as follows:

    R(n)=0.5.sup. n/η!.spsp.4

where η is the characteristic life. Once η is determined, thereliability function is determined. In this case, η characterizes theproduct quality for the reliability requirement R(n) at the operatingcycle 844.

For example, if the reliability goal for solder joints is R(844)=99.9%through the reliability allocation, the characteristic life η isdetermined as follows: ##EQU15## Therefore, for a given confidencelevel, substituting Equations 9 and 11 into Equation 8 gives thefollowing required sample size as a function of the number of thermalcycles: ##EQU16## which results in

    W(n)=8.18×10.sup.14 /n.sup.4 for CL=80%

    W(n)=1.17×10.sub.15 /n.sup.4 for CL=90%

For other reliability goals, the required sample sizes can be determinedby using the same process. FIG. 4 shows the required sample size as afunction of the number of testing cycles for reliability goals of 99.0%,99.9%, 99.97%, and 99.99%, respectively. The solid lines are for theConfidence Level of 90% and the dashed lines are for the ConfidenceLevel of 80%.

The required sample size can be determined for a given reliability goaleither by the above calculations or from the diagram in FIG. 4. Forexample, if the reliability, goal is 99.97% with a confidence level of90%, the solder joints should be tested, without failure, to 2300 cycleswith a sample size of 140. If the sample size is limited to 60, therequired number of test cycles would be 2850 cycles.

EXAMPLE 2 MODULE MOUNTING BRACKETS

A steel bracket may be used to mount an electronic ignition module inthe under-hood area of a passenger car. An accelerated random vibrationtest may be used to qualify the brackets for high-cycle fatigue. Assumethat the field vibration acceleration can be described by the powerspectral density (PSD) function presented in SAE J1211environmentalpractices for electronic equipment. The vibration is applied in thefrequency range from 4.5 Hz to 250 Hz. as shown by the dashed line inFIG. 5. In order to verify the reliability for 100,000 miles, anequivalent duration of six hours is specified at the level shown by thesolid line in FIG. 5. Corresponding to this level the exaggerationfactor is 32.74.

For steel, it is reasonable to assume that the fatigue reliabilityfunction follows the lognormal distribution, that is: ##EQU17## where Φis the standard cumulative normal distribution function, μ_(N) and σ_(N)are the mean value and standard deviation of the logarithm value of thelife. Since the experimental data for this particular material is notavailable, it is conservative to estimate the relationship between μ_(N)σ_(N) and as:

    σ.sub.N ≈0.125 μ.sub.N or δ=12.5%

Therefore, the product quality for the reliability requirement R(n) atthe end of six testing hours can be characterized by the parameterμ_(N).

For example, if the reliability goal of the bracket is R(6)=99.9%, thenthrough the reliability allocation the characteristic life μ_(N) isdetermined as follows: ##EQU18## Therefore, for a given confidencelevel, substituting Equations 14 and 17 into Equation 8 gives thefollowing required sample size as the function of vibration test time:##EQU19##

Similarly, for other values of reliability goals, the required samplesizes can be determined by using the same procedure. FIG. 6 shows therequired sample size vs. testing time for reliability, goals of 99.0%,99.9%, 99.97%, and 99.99%, respectively. The solid lines are with aconfidence level of 90% and the dashed lines are with a confidence levelof 80%. From FIG. 6, the required sample size can be determined normallyfor a given reliability goal. For example, if the reliability goal is99.97% with the confidence level of 90%, then the bracket should betested without failure to 10.5 hours with the sample size of 60. If thesample size is limited to 20, the required testing time is 14 hours.

ACCELERATED TESTING WITH FAILURES

The processes and examples discussed above address the relationshipbetween the required sample size and test cycles/time with no failuresallowed. In many situations, failures do occur during tests. This raisesthe question whether or not the product being tested still meets thereliability goal for the specified test cycles/time when one or morefailures are encountered. This question can be answered by calculatingthe confidence level according to Equation 7. In using Equation 7 todetermine the confidence level, the term R is not the reliability goalas discussed previously, but now become the reliability requirement tototal tested cycles/time.

The process for determining the confidence level of the product(s) beingtested begins with the same steps as previously explained above.

(a) The Reliability Goal (R) is specified for the product, asappropriate, based on the reliability allocation from the systemreliability goal over the product service lifetime. A typicalreliability goal of a product could be 99.99%, but this variable may berange from about 99% to 99.99999%.

(b) The Confidence Level (CL) appropriate for the accuracy of thecalculation/answers required from the accelerated testing is specified.A typical CL is 85%, but this variable may range from about 70% to 95%depending on the critical nature of the application and the importanceof the testing results to the survival of the product being tested.

(c) Either N_(t) (the number of testing cycles or testing duration) or W(the sample size) is selected, based upon the specific requirements inthe case.

(d) Next, the remaining variable, either N_(t) or W (the variable notselected in step (c)), is calculated using the appropriate equation asexplained above (see equations 12 and 13 above).

(e) Testing is initiated N_(t) the specified values of R, CL, N_(t) ,and W, but at least one failure occurs before N_(t) is reached.

(f) If failures are numerous with respect to W (for example, 20%) oroccur early with respect to N_(t) (for example, before 50% N_(t)), thentesting is usually terminated and the root cause of the failure modemust be determined in order to redesign the portion of the product orthe manufacturing process contributing to the failures.

(g) If failures are few and occur relatively late in the testingprogram, then the testing program is completed for N_(t), and W.

(h) If at the completion of the initial testing program for the durationN_(t) and for all W the failures continue to be small with respect to W,then the duration/cycles of the test are extended by a factor of between2 to 5, with the factors 2 or 3 being used in the preferred mode of theprocess. This factor should be toward the higher end of the range if thefailures are numerous compared to W and/or are early with respect toN_(t).

(i) After completion of the extended testing program in the previousstep, the new Confidence Level CL is calculated using equation (4)above: ##EQU20## where x is iterated from .O slashed. to NF=number offailed products, and N_(t) is the extended test duration and W is samplesize.

(j) The calculated Confidence Level CL is compared to a minimumConfidence Level CL_(min) that is determined by considering theresulting effects of the failure mode experienced and the criticalnature of the test results to the survival of the products being tested.For high volume, in high quality automotive component design andmanufacturing applications, CL_(min) should be in the range of 75% to90%, with 80% being preferred. If CL>CL_(min), then the requiredReliability R has been validated. If

CL<CL_(min), then the root cause of the failure mode(s) must bedetermined and the product or manufacturing process corrected to avoidor minimize the failure.

For the case of thermal cycling test of solder joints discussed earlierin Example 1, the sample size can be assumed to be W=200. And for thecase of random vibration test, the sample size can be assumed to beW=30.

It can be calculated from Equation 22 that for a sample size of 200, ifthe reliability goal is 99%, only one failure is allowed within 1000cycles. If two failures occur before 1000 cycles, the testing must becontinued. If the reliability goal is 99.99%, the testing must becontinued to 3000 cycles with only one failure allowed.

Similarly, for the random vibration test of the module mounting bracketsexplained in Example 2 above, it can be calculated from Equation 22 thatif the reliability goal is 99%, no failure is allowed in an eight hourtest. If any failure occurs, the test must be extended to ten hours. Ifthe reliability, goal is 99.95%, testing must be extended to twelvehours and no failures are allowed.

This additional testing beyond the initially calculated acceleratedtesting mimimums must be conducted to confirm if the evaluation basedupon the first failures in the sample group is consistent with theremainder of the testing population.

In summary, the confidence level of an accelerated reliability test isdependent on the reliability requirement, the sample size, and thenumber of allowed failures. The reliability requirement could be alteredby increasing the level of accelerated stress or expanding the testcycles/time. There are three approaches to reduce the sample size for agiven reliability goal and confidence level: (i) increase theacceleration level, (ii) increase test cycles/time, and (iii) decreasethe allowable number of failures. The first approach takes the risk ofshifting the failure mechanism by increasing an already acceleratedstress level. Therefore, the present invention focuses on a method toreduce the sample size based on the second and third approaches. Thepresent invention utilizes a method/process for reducing the test samplesize requirement for the specific reliability goal by expanding the testcycles/time from that correlated to field operational condition.

While the best mode of the present invention has been described indetail, one skilled in this art will be capable of numerous variations,modifications and adaptations without departing from the spirit andscope of the present invention. It should be understood that the presentinvention is not limited to the processes embodiments or examplescontained herein, but are limited only by the scope of the followingclaims.

I claim:
 1. A method for determining a Sample Size W required foraccelerated testing of a product, the method comprising the steps:(a)selecting a Reliability Goal R as appropriate for the product, (b)selecting a Confidence Level CL appropriate for an accuracy requiredfrom the results of the accelerated testing, (c) selecting a number oftesting cycles N_(t) defining the accelerated testing period, (d)calculating the Sample Size W for the accelerated testing as ##EQU21##(e) testing the W product samples for the N_(t) testing cycles tovalidate the required Reliability Goal R when no test failures areobserved over the N_(t) testing cycles.
 2. The accelerated testingmethod as described in claim 1, further comprising the step of:(f)redesigning the product to eliminate the root cause of failures observedin the N_(t) testing cycles.
 3. The accelerated testing method asdescribed in claim 1, further comprising the step of:(f) extending thenumber of testing cycles to at least 2 N_(t) following any failures ofthe products in the Sample Size W, (g) calculating a new ConfidenceLevel as ##EQU22## where NF is a number of failed products within 2N_(t)cycles, (h) validating the Reliability Goal R for the product design ifthe new Confidence Level CL_(NEW) is greater than a CL_(min) valuespecified as a minimum Confidence Level required for the acceleratedtesting method.
 4. The accelerated testing method as described in claim1, further comprising the substep of:(a1) selecting the Reliability GoalR from the range of about 99% to 99.9999% representative of the level ofreliability required from the product.
 5. The accelerated testing methodas described in claim 1, further comprising the substep of:(b1)selecting the Confidence Level CL from the range of about 70% to 95%representative of the Confidence Level required for the acceleratedtesting method.
 6. A method for determining a Number of Cycles N_(t)required for accelerated testing of a product, the method comprising thesteps:(a) selecting a Reliability Goal R as appropriate for the product,(b) selecting a Confidence Level CL appropriate for an accuracy requiredfrom the accelerated testing, (c) selecting the number of productsamples for a Sample Size W, (d) calculating the Number of cycles N_(t)for the accelerated testing as ##EQU23## where R⁻¹ is the inversefunction of R(n), and (e) testing the W product samples for the N_(t)testing cycles to validate the required Reliability Goal R when no testfailures are observed over the N_(t) testing cycles.
 7. The acceleratedtesting method as described in claim 6, further comprising the stepof:(f) redesigning the product to eliminate the root cause of earlyfailures observed in the N_(t) testing cycles.
 8. The acceleratedtesting method as described in claim 6, further comprising the stepof:(f) extending the number of testing cycles to at least 2 N_(t)following any failures of the products in the Sample Size W, (g)calculating a new Confidence Level as ##EQU24## where NF is a number offailed products within 2N_(t) cycles, (h) validating the ReliabilityGoal R for the product design if the new Confidence Level CL is greaterthan a Cl_(min) value specified as a minimum Confidence Level requiredfor the accelerated testing method.
 9. The accelerated testing method asdescribed in claim 6, further comprising the step of:selecting theReliability Goal R from the range of about 99% to 99.9999%representative of the level of reliability required for the product. 10.The accelerated testing method as described in claim 6, furthercomprising the step of:selecting the Confidence Level CL from the rangeof about 70% to 95% representative of the Confidence Level required forthe accelerated testing method.
 11. A method for determining a SampleSize W required for accelerated testing of a product, the methodcomprising the steps:(a) selecting a Reliability Goal R as appropriatefor the product, (b) selecting a Confidence Level CL appropriate for anaccuracy required from the results of the accelerated testing, (c)selecting a number of testing cycles N_(t) defining the acceleratedtesting period, (d) calculating the Sample Size W for the acceleratedtesting as: ##EQU25## (e) testing the W product samples for the N_(t)testing cycles to validate the required Reliability Goal R when no testfailures are observed over the N_(t) testing cycles. (f) extending thenumber of testing cycles to at least 2 N_(t) following any failures ofthe products in the Sample Size W during the N_(t) testing cycles, (g)calculating a new Confidence Level as ##EQU26## where NF is a number offailed products within 2 N_(t) cycles, (h) validating the ReliabilityGoal R for the product design if the new Confidence Level CL is greaterthan a Cl_(min) value specified as a minimum Confidence Level requiredfor the accelerated testing method.
 12. The accelerated testing methodas described in claim 11, further comprising the substep of:(a1)selecting the Reliability Goal R from the range of about 99% to 99.9999%representative of the level of reliability required for the product. 13.The accelerated testing method as described in claim 11, furthercomprising the substep of:(b1) selecting the Confidence Level CL fromthe range of about 80% to 90% representative of the Confidence Levelrequired for the accelerated testing method.